How to solve camera for only part of footage?
It helps if you understand a bit deeper how tracking and 3d reconstruction work.
When you track points in the video and solve the camera you get two different kinds of information: Camera intrinsics and extrinsic parameters.
Camera intrinsics are the camera lens, lens center and distortion. Intrinsics are unique for a combination of sensor size and a specific lens. So all shots done with the same lens on the same camera will behave the same.
Once you have the camera intrinsics, then you can calculate the extrinsics, which are the correspondenc of the placement of points in an image, with coordinates in 3d space.
pff.... got that out.
So what does that mean in this case? You just need to solve a small portion of the scene to determine the camera intrinsics (lens field of vies, optical center and distortion coefficients). With that information you can solve other segments of the shot as well.
At the very least you need 2 images that show parallax and have 8 accurate trackers to solve the scene, but it helps if you have a few more and  more frames to evaluate more precisely.
The segment to use can be defined in the keyframe section, the range of frames between frame A and B will be the one used to solve the scene.

But setting that range doesn't mean that only that segment of the video clip will be solved. Blender will still try to solve the whole thing, but by setting the range means that you don't need 8 trackers on every frame.
And now for the grand finale:
Yes! you need to add trackers for the whole shot if you want to solve it. Don't be lazy or try to take a shortcut: It's only a minute of footage, a limited number of frames and blender is doing most of the work! But remember that not all of the tracking needs to be wonderful, just the segments used between keyframes. Bottom line is: unless you add trackers blender will never guess any motion for you.